In the paper, a system of two linear discrete equations is considered. Numerous sufficient conditions are derived for the existence of solutions to such systems with prescribed asymptotic behavior of solutions. In the literature, not many recommendations are given as to how to detect the initial data generating such solutions. For the case of a system of two linear equations, this gap is filled in this paper. Using the limits of special convergent sequences, sets of initial data are constructed generating the desired solutions.

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